A monad over a categoryMathworldPlanetmath 𝒞 is a triple (T,η,μ), where T is an endofunctor of 𝒞, η is a natural transformation from the identity functor on 𝒞, and μ is a natural transformations from TT to T, such that the following two properties hold:

  • μ(μT)μ(Tμ)

  • μ(Tη)id𝒞μ(ηT)

These laws are illustrated in the following diagrams.